3.44/1.75	YES
3.44/1.76	proof of /export/starexec/sandbox/benchmark/theBenchmark.xml
3.44/1.76	# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty
3.44/1.76	
3.44/1.76	
3.44/1.76	Termination w.r.t. Q of the given QTRS could be proven:
3.44/1.76	
3.44/1.76	(0) QTRS
3.44/1.76	(1) QTRSToCSRProof [SOUND, 0 ms]
3.44/1.76	(2) CSR
3.44/1.76	(3) CSRRRRProof [EQUIVALENT, 31 ms]
3.44/1.76	(4) CSR
3.44/1.76	(5) CSRRRRProof [EQUIVALENT, 0 ms]
3.44/1.76	(6) CSR
3.44/1.76	(7) RisEmptyProof [EQUIVALENT, 0 ms]
3.44/1.76	(8) YES
3.44/1.76	
3.44/1.76	
3.44/1.76	----------------------------------------
3.44/1.76	
3.44/1.76	(0)
3.44/1.76	Obligation:
3.44/1.76	Q restricted rewrite system:
3.44/1.76	The TRS R consists of the following rules:
3.44/1.76	
3.44/1.76	   active(f(b, X, c)) -> mark(f(X, c, X))
3.44/1.76	   active(c) -> mark(b)
3.44/1.76	   active(f(X1, X2, X3)) -> f(X1, active(X2), X3)
3.44/1.76	   f(X1, mark(X2), X3) -> mark(f(X1, X2, X3))
3.44/1.76	   proper(f(X1, X2, X3)) -> f(proper(X1), proper(X2), proper(X3))
3.44/1.76	   proper(b) -> ok(b)
3.44/1.76	   proper(c) -> ok(c)
3.44/1.76	   f(ok(X1), ok(X2), ok(X3)) -> ok(f(X1, X2, X3))
3.44/1.76	   top(mark(X)) -> top(proper(X))
3.44/1.76	   top(ok(X)) -> top(active(X))
3.44/1.76	
3.44/1.76	The set Q consists of the following terms:
3.44/1.76	
3.44/1.76	   active(c)
3.44/1.76	   active(f(x0, x1, x2))
3.44/1.76	   f(x0, mark(x1), x2)
3.44/1.76	   proper(f(x0, x1, x2))
3.44/1.76	   proper(b)
3.44/1.76	   proper(c)
3.44/1.76	   f(ok(x0), ok(x1), ok(x2))
3.44/1.76	   top(mark(x0))
3.44/1.76	   top(ok(x0))
3.44/1.76	
3.44/1.76	
3.44/1.76	----------------------------------------
3.44/1.76	
3.44/1.76	(1) QTRSToCSRProof (SOUND)
3.44/1.76	The following Q TRS is given: Q restricted rewrite system:
3.44/1.76	The TRS R consists of the following rules:
3.44/1.76	
3.44/1.76	   active(f(b, X, c)) -> mark(f(X, c, X))
3.44/1.76	   active(c) -> mark(b)
3.44/1.76	   active(f(X1, X2, X3)) -> f(X1, active(X2), X3)
3.44/1.76	   f(X1, mark(X2), X3) -> mark(f(X1, X2, X3))
3.44/1.76	   proper(f(X1, X2, X3)) -> f(proper(X1), proper(X2), proper(X3))
3.44/1.76	   proper(b) -> ok(b)
3.44/1.76	   proper(c) -> ok(c)
3.44/1.76	   f(ok(X1), ok(X2), ok(X3)) -> ok(f(X1, X2, X3))
3.44/1.76	   top(mark(X)) -> top(proper(X))
3.44/1.76	   top(ok(X)) -> top(active(X))
3.44/1.76	
3.44/1.76	The set Q consists of the following terms:
3.44/1.76	
3.44/1.76	   active(c)
3.44/1.76	   active(f(x0, x1, x2))
3.44/1.76	   f(x0, mark(x1), x2)
3.44/1.76	   proper(f(x0, x1, x2))
3.44/1.76	   proper(b)
3.44/1.76	   proper(c)
3.44/1.76	   f(ok(x0), ok(x1), ok(x2))
3.44/1.76	   top(mark(x0))
3.44/1.76	   top(ok(x0))
3.44/1.76	
3.44/1.76	Special symbols used for the transformation (see [GM04]):
3.44/1.76	top: top_1, active: active_1, mark: mark_1, ok: ok_1, proper: proper_1
3.44/1.76	The replacement map contains the following entries:
3.44/1.76	
3.44/1.76	f: {2}
3.44/1.76	b: empty set
3.44/1.76	c: empty set
3.44/1.76	The QTRS contained just a subset of rules created by the complete Giesl-Middeldorp transformation. Therefore, the inverse transformation is sound, but not necessarily complete.
3.44/1.76	----------------------------------------
3.44/1.76	
3.44/1.76	(2)
3.44/1.76	Obligation:
3.44/1.76	Context-sensitive rewrite system:
3.44/1.76	The TRS R consists of the following rules:
3.44/1.76	
3.44/1.76	   f(b, X, c) -> f(X, c, X)
3.44/1.76	   c -> b
3.44/1.76	
3.44/1.76	The replacement map contains the following entries:
3.44/1.76	
3.44/1.76	f: {2}
3.44/1.76	b: empty set
3.44/1.76	c: empty set
3.44/1.76	
3.44/1.76	----------------------------------------
3.44/1.76	
3.44/1.76	(3) CSRRRRProof (EQUIVALENT)
3.44/1.76	The following CSR is given: Context-sensitive rewrite system:
3.44/1.76	The TRS R consists of the following rules:
3.44/1.76	
3.44/1.76	   f(b, X, c) -> f(X, c, X)
3.44/1.76	   c -> b
3.44/1.76	
3.44/1.76	The replacement map contains the following entries:
3.44/1.76	
3.44/1.76	f: {2}
3.44/1.76	b: empty set
3.44/1.76	c: empty set
3.44/1.76	Used ordering:
3.44/1.76	Polynomial interpretation [POLO]:
3.44/1.76	
3.44/1.76	   POL(b) = 1
3.44/1.76	   POL(c) = 2
3.44/1.76	   POL(f(x_1, x_2, x_3)) = 2*x_2 + 2*x_3
3.44/1.76	With this ordering the following rules can be removed by the rule removal processor [LPAR04] because they are oriented strictly:
3.44/1.76	
3.44/1.76	   c -> b
3.44/1.76	
3.44/1.76	
3.44/1.76	
3.44/1.76	
3.44/1.76	----------------------------------------
3.44/1.76	
3.44/1.76	(4)
3.44/1.76	Obligation:
3.44/1.76	Context-sensitive rewrite system:
3.44/1.76	The TRS R consists of the following rules:
3.44/1.76	
3.44/1.76	   f(b, X, c) -> f(X, c, X)
3.44/1.76	
3.44/1.76	The replacement map contains the following entries:
3.44/1.76	
3.44/1.76	f: {2}
3.44/1.76	b: empty set
3.44/1.76	c: empty set
3.44/1.76	
3.44/1.76	----------------------------------------
3.44/1.76	
3.44/1.76	(5) CSRRRRProof (EQUIVALENT)
3.44/1.76	The following CSR is given: Context-sensitive rewrite system:
3.44/1.76	The TRS R consists of the following rules:
3.44/1.76	
3.44/1.76	   f(b, X, c) -> f(X, c, X)
3.44/1.76	
3.44/1.76	The replacement map contains the following entries:
3.44/1.76	
3.44/1.76	f: {2}
3.44/1.76	b: empty set
3.44/1.76	c: empty set
3.44/1.76	Used ordering:
3.44/1.76	Polynomial interpretation [POLO]:
3.44/1.76	
3.44/1.76	   POL(b) = 1
3.44/1.76	   POL(c) = 0
3.44/1.76	   POL(f(x_1, x_2, x_3)) = x_1 + x_2
3.44/1.76	With this ordering the following rules can be removed by the rule removal processor [LPAR04] because they are oriented strictly:
3.44/1.76	
3.44/1.76	   f(b, X, c) -> f(X, c, X)
3.44/1.76	
3.44/1.76	
3.44/1.76	
3.44/1.76	
3.44/1.76	----------------------------------------
3.44/1.76	
3.44/1.76	(6)
3.44/1.76	Obligation:
3.44/1.76	Context-sensitive rewrite system:
3.44/1.76	R is empty.
3.44/1.76	
3.44/1.76	----------------------------------------
3.44/1.76	
3.44/1.76	(7) RisEmptyProof (EQUIVALENT)
3.44/1.76	The CSR R is empty. Hence, termination is trivially proven.
3.44/1.76	----------------------------------------
3.44/1.76	
3.44/1.76	(8)
3.44/1.76	YES
3.44/1.77	EOF